AN IRISH TRAGEDY vectors and rotations for three-dimensional space were rooted hi his subconscious conviction that the most important laws of common aigebra must persist in the algebra he was seeking. How were vectors in three-dimensional space to be multiplied together? To sense the difficulty of the problem it is essential to bear in mind (see Chapter on Gauss) that ordinary complex numbers a j. oi (z = V — 1) had been given a simple interpretation in terms of rotations in a plane, and further that complex numbers obey all the rules of common algebra, in particular the commuta- tive law of multiplication: it A, B are any complex numbers, then A x B — B x A, whether A, B are interpreted algebraically, or in terms of rotations in a plane. It was but human then to anticipate that the same commutative law would hold for the generalizations of complex numbers which represent rotations in space of three dimensions. Hamilton's great discoverjr - or invention - was an algebra, one of the 'natural' algebras of rotations in space of three dimensions, in which the commutative law of multiplication does not hold. In this Hamiltonian algebra of quaternions (as he called his invention), a multiplication appears in which A ;< B is not equal to B x A but to minus B x A, that is, A y. B = - B x A. That a consistent, practically useful system of algebra could be constructed in defiance of the commutative law of multipli- cation was a discovery of the first order, comparable, perhaps, to the conception of non-Euclidean geometry. Hamilton him- self was so impressed by the magnitude of what suddenly dawned on his mind (after fifteen years of fruitless thought) one day (16 October 1843) when he was out walking with his wife that he carved the fundamental formulae of the new algebra in the .stone of the bridge on which he found himself at the moment. His great invention showed algebraists the way to other algebras until to-day, following Hamilton's lead, mathe- maticians manufacture algebras practically at will by negating one or more of the postulates.for a field and developing tht consequences. Some of these "algebras'1 are extremely useful; the general theories embracing swarms of them include Hamil-